# US 11,441,991 B2 · Multi-angle dynamic light scattering

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- Number: US 11,441,991 B2
- Document ID: US11441991B2
- Status: Granted
- Kind: B2
- Application number: 16/965,204
- Filed: 2019-02-06
- Published: 2021-03-04
- Granted: 2022-09-13
- Earliest claimed: 2018-02-06
- Assignees: Malvern Panalytical Limited
- Applicants: Malvern Panalytical Limited
- Inventors: Douglas Hamilton; Jason Corbett
- Web page: https://patentgrep.com/patent/US11441991B2

## CPC

- G01N 15/0211: {Investigating a scatter or diffraction pattern}
- G01N 2015/0222: {from dynamic light scattering, e.g. photon correlation spectroscopy}
- G01N 2015/025: {Methods for single or grouped particles}
- G01N 2021/4711: {Multiangle measurement}
- G01N: INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES (measuring or testing apparatus or processes other than immunoassay, involving enzymes or microorganisms C12M, C12Q)

## Abstract

A method of determining particle size distribution from multi-angle dynamic light scattering data, comprising: obtaining a series of measured correlation functions g(θi) at scattering angles θi; and solving an equation comprising [ g ⁡ ( θ 1 ) … g ⁡ ( θ n ) ] = [ α 1 ⁢ K ⁡ ( θ 1 ) … α n ⁢ K ⁡ ( θ n ) ] ⁢ x , wherein: K(θi) is the instrument scattering matrix computed for angle i, x is the particle size distribution, and αi is the scaling coefficient for angle i. The method comprises using the steps: a) providing initial estimates for scaling factors α2 to αn, and defining α1=1; b) iterating scaling factors α2 to αn using a non-linear solver; c) solving for x using a linear solver; d) calculate residual; e) repeat steps b) to d) while the residual is greater than a predefined exit tolerance.

## Claims

### Claim 1

A method of determining particle size distribution from multi-angle dynamic light scattering data, comprising:
a light source configured to illuminate a sample on or in a sample holder, wherein the sample comprises a fluid with suspended particles;
a detector configured to detect light scattered by the particles in the sample; and
a processor, wherein the processor is configured to:
determine a series of measured correlation functions g(θi) at scattering angles θi from the scattered light detected by the detector;
solve an equation comprising:
[
g
⁡
(
θ
1
)
…
g
⁡
(
θ
n
)
]
=
[
α
1
⁢
K
⁡
(
θ
1
)
…
α
n
⁢
K
⁡
(
θ
n
)
]
⁢
x
wherein: KW is the instrument scattering matrix computed for angle i, x is the particle size distribution, and α1 is the scaling coefficient for angle i;
a) provide initial estimates for scaling factors a2 to an, and defining a1=1;
b) iterate scaling factors a2 to an using a non-linear solver;
c) solve for x using a linear solver;
d) calculate residual;
e) repeat steps b) to d) while the residual is greater than a predefined exit tolerance;
determine a resulting particle size distribution when the residual is not greater than the predefined exit tolerance; and
display the resulting particle size distribution.

### Claim 2 (depends on 1)

The method of claim 1, wherein the linear solver is NNLS, and/or the non-linear solver is selected from Nelder-Mead simplex, Levenberg-Marquardt and Gauss-Newton.

### Claim 3 (depends on 1)

The method of claim 1, wherein the initial estimates for the scaling factors α2 to αn are estimated by extrapolation of a correlation function to a zero-delay time (τ=0).

### Claim 4 (depends on 1)

The method of claim 1, wherein the predefined exit tolerance is:
a convergence criterion based the preceding residual, or
the predefined exit tolerance is an absolute residual threshold.

### Claim 5 (depends on 1)

The method of claim 1, further comprising repeating the steps a) to e) for a different non-linear solver, to determine which non-linear solver provides the smallest residual.

### Claim 6 (depends on 1)

The method of claim 1, further comprising measuring a time history of scattered light intensity at each respective scattering angle, θ, and determining the correlation functions g(θi) for each scattering angle.

### Claim 7

A method of determining particle size distribution from dynamic light scattering data, comprising:
a light source configured to illuminate a sample on or in a sample holder, wherein the sample comprises a fluid with suspended particles;
a detector configured to detect light scattered by the particles in the sample and determine measurement data from the scattered light; and
a processor, wherein the processor is configured to:
determine, from the scattered light detected by the detector, measured correlation functions g(θ1) to g(θn);
solve an equation of the form:
[
g
⁡
(
θ
1
)
…
g
⁡
(
θ
n
)
]
=
[
α
1
⁢
K
⁡
(
θ
1
)
…
α
n
⁢
K
⁡
(
θ
n
)
]
⁢
x
wherein:
g(θi) is the measured correlation function for measurement time i, corresponding with a scattering angle θi, K(θi) is the instrument scattering matrix computed for the angle θi, x is the particle size distribution and αi is the scaling coefficient for measurement time i (with a1=1);
the vector x takes the form: [x1, xN, n1, . . . , nn], where x1 to xN are the scattering intensities corresponding with each size bin, and the terms n1 to nn are noise intensities corresponding with each of the measurement times or angles; and
columns in K are computed as the expected instrument response according to each element in x, with columns in K at indexes according to elements x1 to xN calculated for each correlator lag-time,τ and angle, θ, and columns at indexes n1 to nn calculated as a computed noise contribution for each correlator lag-time,τ, at each angle, θ, based on an assumption about the characteristics of the noise;
determine a resulting particle size distribution based at least on the computed expected instrument response and the computed noise contribution; and
display the resulting particle size distribution.

### Claim 8 (depends on 7)

The method of claim 7, wherein the dynamic light scattering data is multi-angle data, and each measurement time corresponds with a different measurement angle θi; or
wherein the dynamic light scattering data is of single angle type, and each measurement time i corresponds with the same measurement angle θi.

### Claim 9 (depends on 7)

The method of claim 7, wherein the computed noise contribution is based on the expected instrument response to a large particle in a scattering volume of the instrument; wherein the large particle is assumed to be at least 3 microns in diameter, or at least 10 microns in diameter.

### Claim 10 (depends on 9)

The method of claim 9, wherein the computed noise contribution is determined according to:
g1(τ)=exp(−qi2Dtτ)
where:
g1(τ) is the instrument-measured field autocorrelation functions at lag time, τ; q is the scattering wave vector
(
q
=
4
⁢
π
⁢
⁢
n
o
λ
⁢
sin
⁡
(
θ
i
2
)
)
;
n0 is the dispersant refractive index; λ is the vacuum wavelength; θi is the scattering angle; Dt is the translational diffusion coefficient
(
D
t
=
k
B
⁢
T
3
⁢
⁢
π
⁢
⁢
η
⁢
⁢
d
)
;
kB is the Boltzmann constant; T is the absolute temperature; η is the dispersant viscosity; and d is the assumed large particle hydrodynamic diameter.

### Claim 11 (depends on 7)

The method of claim 7, further comprising sequentially measuring a time history of scattered light intensity at each respective scattering angle, θ, and determining the correlation functions g(θn) for each scattering angle.

### Claim 12

A method of determining a particle size distribution, x, from a dynamic light scattering measurement, comprising:
a light source configured to illuminate a sample on or in a sample holder, wherein the sample comprises a fluid with suspended particles;
a detector configured to detect light scattered by the particles in the sample and measure a time sequence of scattering intensity; and
a processor, wherein the processor is configured to:
determine a measured correlation function, g, derived from the time sequence of scattering intensity;
solve a system of linear equations comprising:
[
g
0
]
=
[
K
γ
⁢
Γ
]
⁢
x
by minimising the sum of residuals:
∥Kx−g∥2+∥γΓx∥2
where K is an instrument scattering matrix, r is a regularisation matrix, and γ is a regularisation vector comprising regularisation coefficients for each particle size in the particle size distribution, x;
determine a resulting particle size distribution based at least on the regularisation vector; and
display the resulting particle size distribution.

### Claim 13 (depends on 12)

The method of claim 12, further comprising performing a measurement to obtain the time sequence of scattering intensity.

### Claim 14 (depends on 12)

The method of claim 12, wherein the system of linear equations to be solved includes regularisation, thereby comprising:
[
g
⁡
(
θ
1
)
…
g
⁡
(
θ
n
)
0
]
=
[
α
1
⁢
K
⁡
(
θ
1
)
…
α
n
⁢
K
⁡
(
θ
n
)
γ
⁢
⁢
Γ
]
⁢
x
optionally wherein:
the vector x takes the form: [x1, . . . , xN, n1, . . . , nn], where x1 to xN are the scattering intensities corresponding with each size bin, and the terms n1 to nn are noise intensities corresponding with each of the measurement times or angles; and
columns in K are computed as the expected instrument response according to each element in x, with columns in K at indexes according to elements x1 to xN calculated for each correlator lag-time,τ and angle, θ, and columns at indexes n1 to nn calculated as a computed noise contribution, at each angle, θ, based on an assumption about the characteristics of the noise.